Because of this inherent sensitivity to the priors, we have to take more care
when choosing priors on the models' parameters when our goal is to compare
models versus estimating parameters.
For example, in Bayesian phylogenetics, it is commonplace to use “uninformative”
priors, some of which are improper (i.e., they do not integrate to one).
The example above demonstrates that if we have informative data, this objective
Bayesian strategy [1]}, [2]} is defensible if our goal is
to infer the posterior distribution of a model;
we are hedging our bets against specifying a prior that concentrates its
probability density outside of where the true value lies, and we can rely on
the informative data to dominate the posterior.
However, this strategy is much harder to justify if our goal is to compare
marginal likelihoods among models.
First of all, models with improper priors do not have a well-defined marginal
likelihood and should not be used when comparing models [3]}.
Second, even if diffuse priors are proper, they could potentially sink the
marginal likelihood of good models by placing excessive weight in
biologically unrealistic regions of parameter space with low likelihood.
Thus, if our goal is to leverage Bayesian model choice to learn about the
processes that gave rise to our data, a different strategy is called for.
One option is to take a more subjective Bayesian approach
[4]}, [5]}, [6]} by carefully choosing prior
distributions for the models' parameters based on existing knowledge.
In the era of “big data,” one could also use a portion of their data to
inform the priors, and the rest of the data for inference.
Alternatively, we can use hierarchical models that allow the data to inform the
priors on the parameters [7]}.
